According to improvement of a processing performance of computers and a proliferation of inexpensive cameras, a method of recognition by inputting a plurality of images retrieved from a video sequence, obtaining a pattern distribution of the input images and a pattern distribution of dictionary images, and comparing a similarity between the distributions is now in a practical stage in the field of the pattern recognition such as face recognition.
As one of methods of measuring a similarity between the distributions described above, Mutual Subspace Method is proposed in JP-A-2003-248826 (KOKAI). The Mutual Subspace Method recognizes a category of input patterns by approximating input patterns and dictionary patterns in subspaces respectively and measuring the similarity between the subspaces by a canonical angle.
A subspace method in the related art is configured to measure the similarity between one input pattern vector and a subspace which indicates a focused category. In contrast, the Mutual Subspace Method described above is configured to convert an input image sequence into a subspace as well when a plurality of images are input from a target of recognition such as the moving image and measure the similarity among the subspaces. At this time, a square of a cosine of an angle referred to as a canonical angle formed between the two subspaces to be compared is used as the “similarity”.
Bases of two subspaces Sm and Sn are expressed as φ(m)=(φ1(m), . . . , φK(m)) and φ(n)=(φ1(n),. . . , φL(n)) respectively, and it is assumed to be K≦L. The canonical angles are defined by a smaller number of dimensions between the two subspaces (K, in this case), and when ordering the canonical angles in ascending number order as; {θ1(nm), . . . , θK(nm)}, an ith eigenvalue of an eigenvalue problem indicated by an expression (1) shown below is given to a cosθi(nm).Φ(n)TΦ(m)Φ(m)TΦ(n)α=λα  (1)
The eigenvalue problem is introduced as shown below using Lagrange Multiplier Method.
A given unit vector on the subspace Sn is expressed by Φ(n) a by a linear combination of φ(n)i using an Lth dimensional unit vector a (aTa=1). A square of a projecting length when it is projected in the subspace Sm is a TΦ(n)TΦ(m)Φ(m)TΦ(n)a. Then, in order to maximize the above-described value under the restriction of aTa=1, an expression (2) shown below using a multiplier λ is differentiated with respect to a nd λ. Then, the eigenvalue problem of the expression (1) is obtained with respect to a from the differentiation.αTΦ(n)TΦ(m)Φ(m)TΦ(n)α−λ(αTα−1)   (2)
In the expression (2), a first canonical angle means an angle formed between a unit vector selected from unit vectors in one of the subspaces which achieves a longest projecting length to the other subspace and a vector of the same projected to the other subspace, and is an angle at which the two subspaces get nearest to each other.
The canonical angle used in the Mutual Subspace Method is an angle formed between the two subspaces, and what is meant thereby is clear in terms of geometrical point of view. However, when it is used as the similarity between the patterns, a small conical angle does not necessarily mean that they are similar patterns. The reason is that a criterion of the similarity depends on the problem, and an adequate similarity measure should be learned depending on the problem.
Under an awareness of the issues as described above, Orthogonal Mutual Subspace Method is proposed in Japanese Patent Application No. 2005-035300. This Orthogonal Mutual Subspace Method is configured to obtain a linear transformation matrix which increases the canonical angle between the subspaces which belong to different categories by solving the eigenvalue problem.
In other words, the Orthogonal Mutual Subspace Method obtains an autocorrelation matrix S of basis vectors φim of all of the dictionary subspaces from an expression S=ΣmΣiφimφimt/(ΣmΣi1), and the similarity is obtained by linearly transforming the subspaces by a matrix O obtained by using an expression (3) shown below,O=VΛ−1/2Vt   (3)where V is a matrix of eigenvectors of S and Λ is a diagonal matrix having the eigenvalues of S as diagonal components.
Though the Orthogonal Mutual Subspace method increases the canonical angles between the subspaces witch belong to different categories, it does not necessarily reduce the canonical angles between the subspaces which belong to the same category.
According to an experiment conducted by an inventor, it was found that the Orthogonal Mutual Subspace Method increased not only the canonical angle between the subspaces which belong to the different categories, but also the canonical angle between the subspaces which belong to the same category when being compared with the Mutual Subspace Method in which the linear transformation is not performed. However, since the amount of increase of the canonical angle is larger in the case of the canonical angle between the subspaces which belong to the different categories, a higher performance is expected in comparison with the Mutual Subspace Method. Therefore, the Orthogonal Mutual Subspace Method has a certain advantage over the Mutual Subspace Method from a practical standpoint.
The reason why the result of experiment was as described above will be described below. The Orthogonal Mutual Subspace Method is a method based on a finding such that the canonical angle between the different categories increases when the linear transformation matrix given by the expression (3) is used, and is not a method which is able to operate the canonical angle by itself directly as a target of optimization.
In contrast, in order to improve an accuracy of the pattern recognition, it is more effective to selectively increase the canonical angles of pairs of the subspaces whose canonical angles are small even though they belong to the different categories than to uniformly increase the canonical angle between the subspaces which belong to the different categories since the probability of confusion of the patterns which are easily confused is reduced.
In the same manner, as regards the canonical angle between the subspaces which belong to the same category, it is preferable to selectively reduce the canonical angle of the combination of the subspaces which forms a large canonical angle rather than to reduce the same on an average.
A framework to provide a desirable output value to a combination of the subspaces to be input as a desired output and optimize a parameter so as to minimize a deviation between the teacher signal and an output is generally used in a problem of machine leaning.
However, in the method using the Mutual Subspace Method, it is difficult to perform the learning applying the optimization framework. It is because the basis of the linearly transformed subspace is needed to be converted to an orthonormal basis by Gram-Schmidt orthogonalization, and due to the necessity of this operation when linearly transforming the subspace to obtain the canonical angle, differentiation of the canonical angle with respect to the linear transformation cannot be obtained in a general form.
In view of such problems, it is an object of the invention to provide a linear transformation matrix calculating apparatus in which a linear transformation matrix which decreases similarities among subspaces which belong to different categories, while increasing the similarities between subspaces which belong to the same category is obtained, a method thereof, and a program thereof.